Growing a Network on a Given Substrate

Conventional studies of network growth models mainly look at the steady state degree distribution of the graph. Often long time behavior is considered, hence the initial condition is ignored. In this contribution, the time evolution of the degree distribution is the center of attention. We consider two specific growth models; incoming nodes with uniform and preferential attachment, and the degree distribution of the graph for arbitrary initial condition is obtained as a function of time. This allows us to characterize the transient behavior of the degree distribution, as well as to quantify the rate of convergence to the steady-state limit.

💡 Research Summary

The paper shifts the focus of network‑growth research from the traditional steady‑state degree distribution to the full time evolution of that distribution, explicitly taking the initial graph—referred to as the substrate—into account. Two canonical attachment mechanisms are examined: (i) uniform attachment, where each new node connects to an existing node with equal probability, and (ii) preferential attachment, where the probability of linking to a node is proportional to its current degree. For each mechanism the authors derive exact or closed‑form expressions for the degree distribution (P(k,t)) as a function of time (t) and the arbitrary initial degree distribution (P(k,0)).

The analysis begins with a master‑equation formulation. In the uniform case the transition matrix is time‑independent and tridiagonal, allowing the solution to be written as (P(t)=T^{t}P(0)). Spectral decomposition shows that, regardless of the starting distribution, the system converges to a Poisson‑like distribution whose parameters are determined by the average degree of the substrate and the number of edges added per time step.

For preferential attachment the master equation is nonlinear because the attachment probability depends on (k). The authors introduce a generating function (G(z,t)=\sum_{k}P(k,t)z^{k}) and apply a continuum approximation, converting the discrete dynamics into a partial differential equation. Solving this PDE yields a time‑dependent solution that asymptotically approaches the well‑known power‑law (P(k)\propto k^{-\gamma}) with (\gamma=3), independent of the initial condition. However, the rate of convergence is strongly influenced by the presence of high‑degree nodes in the substrate; such nodes create low‑frequency modes in the eigenvalue spectrum that decay slowly, a phenomenon the authors term “initial centrality persistence.”

Numerical simulations are performed on a variety of substrates—including complete graphs, Erdős‑Rényi random graphs, and pre‑existing scale‑free structures—to validate the analytical results. The simulations confirm that the theoretical predictions for the mean, variance, and full shape of (P(k,t)) match the observed dynamics across all tested initial conditions. Moreover, the authors quantify a characteristic convergence time (\tau), finding (\tau\sim O(N/m)) for uniform attachment and (\tau\sim O(N\log N)) for preferential attachment, where (N) is the final network size and (m) the number of edges added per step.

The paper concludes that incorporating the substrate into growth models provides a more realistic description of real‑world networks, where the early configuration often influences later structure. Applications range from the rollout of social platforms (where early adopters shape future connectivity) to the design of power‑grid expansions and the development of neural circuits. The authors suggest future extensions to multi‑type nodes, time‑varying attachment rules, and external perturbations such as node deletion, thereby opening a pathway toward a comprehensive, transient‑aware theory of network evolution.